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In statistics, high-leverage points are those that are outliers with respect to the independent variables. Leverage points are those that cause large changes in the parameter estimates when they are deleted. Although an inflential point will typically have high leverage, a high leverage point is not necessarily an influential point. The leverage is typically defined as the diagonal of the hat matrix

H = X (X^{'} X)^{- 1} X^{'} .

Partial leverage is used to measure the contribution of the individual independent variables to the leverage of each observation. That is, if h_i is the i^th row of the diagonal of the hat matrix, the partial leverage is a measure of how h_i changes as a variable is added to the regression model.

The partial leverage is computed as:

{({P L}_{j})}_{i} = \frac{{(X_{j ∙ [j]})}_{i}^{2}}{\sum_{k = 1}^{n} {(X_{j ∙ [j]})}_{k}^{2}}

where

j = index of independent variable

i = index of observation

X_j·[j] = residuals from regressing X_j against the remaining independent variables

Note that the partial leverage is the leverage of the i^th point in the partial regression plot for the j^th variable. Data points with large partial leverage for an independent variable can exert undue influence on the selection of that variable in automatic regression model building procedures.

External links

Partial Leverage Plot

References

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20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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-{{TOCright}}
+In [[statistics]], high-leverage points are those that are [[outlier]]s with respect to the [[independent variables]]. [[leverage (statistics)|Leverage point]]s are those that cause large changes in the parameter estimates when they are deleted. Although an inflential point will typically have high leverage, a high leverage point is not necessarily an influential point.  The [[leverage (statistics)|leverage]] is typically defined as the diagonal of the [[hat matrix]]
-'''Gossen's Second “Law”''', named for [[Hermann Heinrich Gossen]] (1810–1858), is the assertion that an [[Economics|economic]] agent will allocate his or her expenditures such that the ratio of the [[marginal utility]] of each [[Good (economics)|good]] or [[Service (economics)|service]] to its price (the marginal expenditure necessary for its acquisition) is equal to that for every other good or service.  Formally,
-:<math>\frac{\partial U/\partial x_i}{p_i}=\frac{\partial U/\partial x_j}{p_j}~\forall\left(i,j\right)</math>
+:<math> H = X(X'X)^{-1}X'.  \, </math>
-where
-* <math>U</math> is [[utility]]
-* <math>x_i</math> is quantity of the <math>i</math>-th good or service
-* <math>p_i</math> is the price of the <math>i</math>-th good or service
-== Informal derivation ==
+'''Partial leverage''' is used to measure the contribution of the individual independent variables to the leverage of each observation. That is, if ''h''<sub>''i''</sub> is the ''i''<sup>th</sup> row of the diagonal of the hat matrix, the partial leverage is a measure of how ''h''<sub>''i''</sub> changes as a variable is added to the regression model.
-Imagine that an agent has spent money on various sorts of goods or services.  If the last unit of currency spent on goods or services of one sort bought a quantity with <em>less</em> marginal utility than that which would have been associated with the quantity of another sort that could have been bought with the money, then the agent would have been ''better off'' instead buying more of that other good or service.  Assuming that goods and services are continuously divisible, the only way that it is possible that the marginal expenditure on one good or service should not yield more utility than the marginal expenditure on the other (or ''vice versa'') is if the marginal expenditures yield ''equal'' utility.
+The partial leverage is computed as:
-== Formal derivation ==
+:<math>
+\left(\mathrm{PL}_j\right)_i = \frac{\left(X_{j\bullet[j]}\right)_i^2}{\sum_{k=1}^n\left(X_{j\bullet[j]}\right)_k^2}
+</math>
-Assume that utility, goods, and services have the requisite properties so that <math>\partial U/\partial x_i</math> is [[Well-defined|well defined]] for each good or service.  An agent then optimizes
-:<math>U\left(x_1 ,x_2 ,\dots,x_n\right)</math>
-subject to a [[budget constraint]]
-:<math>W\geq\sum_{i=1}^n \left(p_i\cdot x_i \right)</math>
 where
-* <math>W</math> is the total available sum of money
+:''j'' = index of independent variable
-Using the method of [[Lagrange multipliers]], one constructs the function
+:''i'' = index of observation
-:<math>\mathcal{L}\left(x_1 ,x_2 ,\dots,x_n ,\lambda\right)=U\left(x_1 ,x_2 ,\dots,x_n\right)+\lambda\cdot\left[W-\sum_{i=1}^n \left(p_i\cdot x_i \right)\right]</math>
+:''X''<sub>''j''&middot;[''j'']</sub> = [[errors and residuals in statistics|residuals]] from regressing ''X''<sub>''j''</sub> against the remaining independent variables
-and finds the first-order conditions for optimization as
-:<math>\frac{\partial\mathcal{L}}{\partial\lambda}=0</math>
-(which simply implies that all of <math>W</math> will be spent) and
-:<math>\frac{\partial\mathcal{L}}{\partial x_i}=0~~\forall i</math>
-so that
-:<math>\frac{\partial U}{\partial x_i}-\lambda\cdot p_i =0~~\forall i</math>
-which is algebraically equivalent to
-:<math>\frac{\partial U/\partial x_i}{p_i}=\lambda~~\forall i</math>
-Since every such ratio is equal to <math>\lambda</math>, the ratios are all equal one to another:
-:<math>\frac{\partial U/\partial x_i}{p_i}=\frac{\partial U/\partial x_j}{p_j}~\forall\left(i,j\right)</math>
-(Note that, as with any maximization using first-order conditions, the equations will hold only if the utility function satisfies specific concavity requirements and does not have maxima on the edges of the set over which one is maximizing.)
-== References ==
+Note that the partial leverage is the leverage of the ''i''<sup>th</sup> point in the [[partial regression plot]] for the ''j''<sup>th</sup> variable. Data points with large partial leverage for an independent variable can exert undue influence on the selection of that variable in automatic regression model building procedures.
-* Gossen, Hermann Heinrich; ''Die Entwicklung der Gesetze des menschlichen Verkehrs und der daraus fließenden Regeln für menschliches Handeln'' (1854).  Translated into English as ''The Laws of Human Relations and the Rules of Human Action Derived Therefrom'' (1983) MIT Press, ISBN 0-262-07090-1.
+==See also==
+* [[Partial residual plot]]
+* [[Partial regression plot]]
+* [[Variance inflation factor]] for a multi-linear fit
+* [[Scatterplot matrix]]
-== See also ==
+==External links==
+* [http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/partleve.htm Partial Leverage Plot]
-* [[Gossen's laws]]
+==References==
-* [[Hermann Heinrich Gossen]]
+* {{cite book
-* [[Marginal utility]]
+|title = Modern Regression Methods
-* [[Marginalism]]
+|author = Tom Ryan
+|publisher = John Wiley
+|year = 1997}}
+* {{cite book
+|title = Applied Linear Statistical Models
+|edition = 3rd
+|author = Neter, Wasserman, and Kunter
+|year = 1990
+|publisher = Irwin}}
+* {{cite book
+|title = Applied Regression Analysis
+|edition = 3rd
+|author = Draper and Smith
+|publisher = John Wiley
+|year = 1998}}
+* {{cite book
+|title = Residuals and Influence in Regression
+|author = Cook and Weisberg
+|publisher = Chapman and Hall
+|year = 1982}}
+* {{cite book
+|title = Regression Diagnostics
+|author = Belsley, Kuh, and Welsch
+|publisher = John Wiley
+|year = 1980}}
+* {{cite journal
+|title = Efficient Computing of Regression Diagnostiocs
+|author = Paul Velleman
+|coauthor = Roy Welsch
+|journal = The American Statistician
+|month = November
+|year = 1981
+|volume = 35
+|pages = 234–242
+|doi = 10.2307/2683296
+|issue = 4
+|publisher = American Statistical Association
+|jstor = 2683296}}
-{{DEFAULTSORT:Gossen's Second Law}}
+{{NIST-PD}}
-[[Category:Marginal concepts]]
-[[Category:Consumer theory]]
-[[Category:Utility]]
-[[nl:Tweede wet van Gossen]]
+[[Category:Regression diagnostics]]

Main Page: Difference between revisions

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